Hedge Funds puzzles, solved step by step

Puzzles: 100
Traced to a firm: 38
Topics: 14
Hard: 30

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014A corporate bond has a spread duration of 6 and convexity of 50. Its credit spread widens by 50 basis points. Roughly what happens to its price?Valuation, accounting and macro riddlesCoreAQR Capital ManagementGreenwich · 2021
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Which is closest?
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The price falls by about 2.94%. Spread duration of 6 says a 0.50 percentage point widening costs 6 x 0.50% = 3.00%. Convexity of 50 adds back one half x 50 x 0.005 squared, about 0.06%, because the price curve bends upwards. On a bond priced at 100 that is a move to about 97.06. At 50 basis points the convexity term is small; at 300 or 500 it is not.
What do duration and convexity each measure?
Picture a playground slide that curves and flattens towards the bottom. Judge the drop from the steepness at the top and you overstate it, because the slide levels off as you go. Spread durationThe percentage change in a bond price for a one percentage point change in its credit spread, holding the risk-free rate fixed. is the steepness at today's spread; convexity is the flattening, so the straight-line estimate always overstates the loss when spreads widen. Duration gives the first-order move, 6 x 0.50% = 3.00% down; convexity corrects it by a term that depends on the square of the move.
For a 50 basis point widening, duration of 6 gives minus 3.00% and convexity of 50 adds back 0.06%, a fall of 2.94%; the convexity cushion grows with the square of the move, to 2.25 points at 300 basis points and 6.25 at 500.
The relationship
$\frac{\Delta P}{P} \approx -D_s\,\Delta s + \tfrac{1}{2}C\,(\Delta s)^2 = -6(0.005) + \tfrac{1}{2}(50)(0.005)^2 = -3.00\% + 0.0625\% \approx -2.94\%$
D_s spread duration, 6
C convexity, 50
\Delta s the change in spread as a decimal, 50 basis points = 0.005
What it says in wordsThe price moves by the duration term plus a smaller correction that grows with the square of the spread change.
When does the convexity term start to matter?
It grows with the square of the move. At 50 basis points convexity is worth 0.06% against a 3.00% duration loss; at 300 basis points duration says -18% and convexity adds back 2.25%, which is no longer small. That is why a credit desk can run duration-only risk for everyday moves but needs convexity for stress scenarios. One more distinction marks a strong answer: for a fixed-coupon bond spread duration and rate duration are close, but a floating-rate note has almost no rate duration and still carries several years of spread duration.
Say the limitation plainly. Both numbers are local, measured at today's spread, and a distressed bond stops behaving like this long before default, when its price starts tracking the expected recovery instead. For a bond trading near par, as here, the two-term estimate is good to a few hundredths of a per cent for moves of this size.
Where candidates lose it
Candidates give minus 3% and stop, which is fine as a first line but ignores the second number the question handed you. Worse is using convexity with the wrong sign, making the loss bigger: for a plain bond convexity always cushions a spread widening.
The other slip is units. Fifty basis points is 0.005 in the formula; squaring 0.50 instead turns a 0.06% correction into 6.25% and produces a price that rises when spreads widen.
What the interviewer asks next
What if the spread tightens by 50 basis points instead?
Why can a callable bond have negative convexity?
How would you hedge the spread risk of this bond?
Asked at AQR Capital Management, Investment Research, Greenwich, 2021 (Wall Street Oasis): Discussion on credit spreads on fixed income products and duration.
049You short a stock at Rs 100 at 5% of NAV. It rises to Rs 150 while the rest of the book is flat. What is your loss, and what share of NAV is the short position now?Valuation, accounting and macro riddlesCoreLong-short equity fundsGlobal macro funds
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The short's share of NAV at Rs 150 is about
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You lose 2.5% of NAV, and the short is now about 7.7% of the book. On NAV of 100, the short is worth 5. A 50% rise makes it a liability of 7.5, a loss of 2.5, so NAV falls to 97.5. The position is now 7.5 over 97.5, which is 7.7%. A losing short grows as a share of the book, so risk rises exactly when the trade is going wrong, the opposite of a losing long.
How do the loss and the new weight work out?
Keep NAV at 100 so every number is a percentage. The short is a liability that rises with the price: 5 at Rs 100 becomes 7.5 at Rs 150, a loss of 2.5, which takes NAV from 100 to 97.5. The weight is the liability over the new NAV: 7.5 / 97.5 = 7.69%. Both parts of the fraction move against you: the top grows and the bottom shrinks.
Why is a losing short more dangerous than a losing long?
A losing long is like a debt that shrinks as the thing you own loses value: a 5% long falling to half is worth 2.5 of NAV 97.5, 2.6%, and it can never lose more than the 5 you put in. A losing short does the reverse: the more it loses, the bigger it gets, and there is no ceiling on the price, so there is no cap on the loss. Left alone, a short that doubles is 10 of NAV 95, 10.5%, and one that triples is 16.7%. This is why long-short funds size shorts smaller and cut them faster than longs.
A 5% long whose price halves shrinks to 2.6% of the book and can lose at most 5% of NAV, while a 5% short whose price rises to 150 grows to 7.7% and keeps growing, with no cap on the loss.
The relationship
$w_{\text{short}} = \frac{0.05 \times 1.5}{1 - 0.05 \times 0.5} = \frac{7.5}{97.5} \approx 7.7\%$
0.05 the starting weight, 5% of NAV
1.5 the price relative, 150 over 100
0.05 x 0.5 the loss as a share of starting NAV, 2.5%
What it says in wordsThe short's new weight is its grown liability over the NAV that the loss has reduced.
Add what a desk does about it. Many funds set a stop or a maximum weight for each short and cover part of it as it rises, so that a losing short is brought back towards its original risk. The discipline exists because the position sizes itself up without anyone deciding to. Short squeezes make this worse, since the price can gap up on little news when many holders cover at once.
Where candidates lose it
The common loss is saying the weight is 7.5%, forgetting that NAV fell. It is 7.5 over 97.5, not over 100. A smaller trap is saying 5%, as if the short's size were fixed at the entry value.
The bigger miss is not drawing the lesson. The interviewer wants to hear that shorts grow when they lose and longs shrink, so short risk compounds against you and needs its own limits.
What the interviewer asks next
At what price does the short reach 10% of NAV?
How much would you buy back to bring the short to 5% of NAV at Rs 150?
The stock pays a dividend while you are short. What happens to your P&L?
064An exporter earns Rs 100 of revenue, 60% of it billed in dollars, and has Rs 85 of costs, all paid in rupees. The rupee weakens from 80 to 88 to the dollar. What happens to its operating margin?Valuation, accounting and macro riddlesCoreLong-short equity fundsGlobal macro funds
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What is the operating margin after the move?
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The margin rises from 15% to about 19.8%, and operating profit rises 40%. Each dollar now buys 10% more rupees, which turns Rs 60 of dollar revenue into Rs 66, so revenue reaches Rs 106. Costs are all in rupees and stay at Rs 85, so profit climbs from Rs 15 to Rs 21, and 21 over 106 is 19.8%. A 6% rise in revenue becomes a 40% rise in profit.
Why does a weaker rupee help this company so much?
Think of a tutor in Pune who teaches some students abroad for fees in dollars and pays rent and a salary in rupees. When the rupee weakens, each dollar fee converts into more rupees while the rent does not move. A currency move lands on the revenue billed in the foreign currency and on nothing else, so when every cost is local the whole gain drops into profit. Here the gain is Rs 6, on a profit that started at only Rs 15.
Before the move, Rs 40 of rupee revenue and Rs 60 of dollar revenue less Rs 85 of rupee costs leave Rs 15 of profit, a 15% margin; at 88 to the dollar the dollar revenue is worth Rs 66, costs stay at Rs 85 and profit rises to Rs 21, a 19.8% margin.
How do you size the profit move quickly?
Chain three numbers. The dollar buys 10% more rupees, 60% of revenue is exposed, so revenue rises 6%; a thin 15% margin turns that Rs 6 into a 40% rise in profit. The ratio of the profit change to the revenue change, 40 over 6, almost 7 times, is the company's operating leverageHow much operating profit moves for a given move in revenue, high when most costs are fixed or unaffected by the change. to the currency. An analyst on a long-short book uses that multiplier to judge how much of a stock's move a currency view is really driving.
The relationship
$\text{margin} = \frac{40 + 60 \times \tfrac{88}{80} - 85}{40 + 60 \times \tfrac{88}{80}} = \frac{21}{106} \approx 19.8\%$
40 revenue billed in rupees, unchanged
60 x 88/80 dollar revenue restated at the new rate, Rs 66
85 costs, all in rupees, unchanged
What it says in wordsRestate only the dollar-billed revenue at the new rate, hold rupee costs fixed, and divide profit by the new revenue.
What would you check before trusting the new margin?
Three things that often shrink the gain. Hedges, pricing and imported inputs: a company that sold its dollars forward keeps the old rate until the hedges roll off, buyers abroad may push for lower dollar prices, and any imported materials get dearer too. Each one eats into the Rs 6. The 19.8% is the unhedged, first-round answer; naming those three tells the interviewer you would not stop at it.
Where candidates lose it
The first slip is saying the margin is unchanged because revenue and costs both grow. Only the dollar-billed revenue moves; rupee costs stay where they were, and that asymmetry is the whole point.
The second is applying the 10% to all Rs 100 of revenue, which gives Rs 110 and a 22.7% margin. Only 60% of revenue is billed in dollars, so revenue rises 6%, not 10%.
What the interviewer asks next
Half of the costs are imported and paid in dollars. What is the new margin?
The company hedged half its dollar revenue at 80. What margin does it report this year?
What does the same move do to an importer with the mirror-image structure?
074A company has a market value of Rs 1,000 crore, earnings of Rs 100 crore, and Rs 300 crore of net cash that earns 6% after tax. What is the P/E of the operating business on its own?Valuation, accounting and macro riddlesCoreLong-short equity fundsGlobal macro funds
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What is the P/E of the operating business alone?
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About 8.5x, not 10x. Rs 300 crore of the Rs 1,000 crore market value is cash, so the operating business is valued at Rs 700 crore. The cash earns 6% after tax, Rs 18 crore, so the operating business earns Rs 82 crore. Rs 700 crore over Rs 82 crore is about 8.5 times earnings. The headline 10x blends a cheaper business with cash priced at about 16.7 times its income.
Why is the headline P/E misleading here?
Imagine buying a shop for Rs 10 lakh that has Rs 3 lakh sitting in its till. You are really paying Rs 7 lakh for the shop itself. A company's price and earnings both include its cash, and cash is valued very differently from a business: here it earns 6% after tax, so it is worth about 16.7 times its income. Blending the two gives the headline 10x, which is neither the price of the cash nor the price of the business.
Removing Rs 300 crore of cash from the Rs 1,000 crore market value leaves Rs 700 crore for the operating business, and removing the Rs 18 crore it earns leaves Rs 82 crore of operating earnings, a multiple of about 8.5x against the headline 10x.
How do you work it?
Strip the cash out of both the numerator and the denominator. The price of the operating business is 1,000 minus 300, Rs 700 crore; its earnings are 100 minus 6% of 300, Rs 82 crore; and 700 over 82 is 8.54x. Taking the cash out of only the price gives 7.0x, which is too low because the Rs 100 crore still contains the cash's interest. The check: 700 plus 300 is 1,000 and 82 plus 18 is 100.
The relationship
$\text{P/E}_{\text{ops}} = \frac{1{,}000 - 300}{100 - 0.06 \times 300} = \frac{700}{82} \approx 8.5\times$
1,000 market value, Rs crore
300 net cash, Rs crore
0.06 x 300 the after-tax income the cash earns, Rs 18 crore
What it says in wordsTake the cash out of the price and its income out of the earnings, then divide.
When would you not strip out all the cash?
When the cash is not really free. Cash needed to run the business, cash trapped abroad that would be taxed on the way home, or cash a management team is likely to spend badly may deserve less than full value. A long-short analyst uses the ex-cash multiple to compare this business with peers that hold no cash, and then asks how much of the Rs 300 crore shareholders will actually see. That judgement moves the answer between about 8.5x and the headline 10x.
Where candidates lose it
The first slip is quoting 10x and moving on, which treats the cash as if it were part of the operating business. The interviewer put Rs 300 crore of cash in precisely so that the headline multiple would mislead.
The second is subtracting the cash from the price but leaving its interest in the earnings, which gives 7.0x. Cash must come out of both sides.
What the interviewer asks next
The cash earns only 3% after tax. What is the operating P/E now?
The company holds Rs 300 crore of net debt instead, at 6% after tax. What is the operating P/E?
Why might the market value the company's cash at less than Rs 300 crore?
089The equity risk premium is 4.5%, and a market's fair multiple is 1 divided by (real yield + premium - real growth). Real yields rise from 1.5% to 2.5% while expected real growth rises from 2.0% to 2.5%. What happens to the fair multiple?Valuation, accounting and macro riddlesCoreCitadelNew York · 2026
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Where does the fair multiple go?
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The fair multiple falls from 25x to about 22.2x, a compression of about 11%. The denominator is the real yield plus the premium minus growth: 1.5 + 4.5 - 2.0 = 4.0% before, and 2.5 + 4.5 - 2.5 = 4.5% after. Real yields rose by a full point and growth by only half a point, so the net rate rose by half a point, and the multiple, its inverse, fell.
Why is the multiple one over a spread?
Think of a shop that pays you rent forever, rising a little each year. What you would pay for it depends on the return you demand minus how fast the rent grows. For earnings paid out and growing forever, price over earnings is one divided by the required return minus growth, so the multiple depends only on the gap between the two. In real terms the required return is the real yield plus the equity risk premiumThe extra return investors demand for holding shares rather than government bonds.. The question treats all earnings as paid out, which is the assumption to name.
The net rate in the denominator rises from 4.0% to 4.5% because real yields climbed a full point while growth climbed half a point, so the fair multiple falls from 25x to 22.2x, about 11% lower.
How do you work it out quickly?
Compute the denominator before and after. Before: 1.5 + 4.5 - 2.0 = 4.0%, a multiple of 25x. After: 2.5 + 4.5 - 2.5 = 4.5%, a multiple of 22.2x. Rates went up by 1.0 point and growth by 0.5, so the spread widened by 0.5 point. A 0.5-point rise on a 4.0% base is a 12.5% rise in the denominator, and the multiple falls by 1 minus 1/1.125, about 11.1%.
The relationship
$\frac{P}{E} = \frac{1}{r + \text{ERP} - g} \qquad \frac{1}{4.0\%} = 25\times \qquad \frac{1}{4.5\%} = 22.2\times$
r the real yield on government bonds
ERP the equity risk premium, 4.5%
g expected real growth of earnings
What it says in wordsThe fair multiple is one over the net rate: what investors demand minus how fast the earnings grow.
What does this teach beyond the arithmetic?
Higher yields do not hurt equities one for one if growth rises with them. What matters is whether real yields rise faster or slower than expected growth: faster compresses multiples, slower expands them. Had growth also risen a full point, to 3.0%, the net rate would be 4.0% again and the multiple 25x. The limitation to state is sensitivity: near a 4% net rate, a half-point move shifts the multiple by about 2.8 turns one way and 3.6 the other, so small errors in the premium or the growth guess swamp the answer.
Where candidates lose it
The quick wrong answer is that nothing happens because both rates went up. The question is built so that growth rises by only half as much as yields, and it is the spread, not the level, that sets the multiple.
The second loss is dropping the growth change and answering 20x. Write the denominator out in full, before and after; it takes ten seconds and removes both errors.
What the interviewer asks next
Growth rises by a full point, to 3.0%. What is the multiple now?
The equity risk premium also falls to 4.0%. What is the net effect?
Why do shares whose value sits far in the future fall more than the market when real yields rise?
Asked at Citadel, Software, New York, 2026 (Wall Street Oasis): real yields rising faster than growth expectations predicts equity multiple compression
099You borrow in yen at 1% and invest in rupee deposits at 7% for a year. By how much can the rupee's value in yen fall over the year before the trade loses money?Valuation, accounting and macro riddlesCoreLong-short equity fundsGlobal macro funds
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How big a fall in the rupee can the trade absorb?
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About 5.6%. Borrow 1 of yen value, convert it, and the rupee deposit grows to 1.07 in yen terms at today's rate. You owe 1.01. If the rupee's yen value falls by d, the deposit is worth 1.07 x (1 - d), which equals 1.01 when d = 1 - 1.01/1.07, about 5.6%. The cushion is slightly below the 6-point gap because the fall hits the interest as well as the principal.
Where does the carry trade's income come from?
Think of borrowing from a relative at 1% to put the money in a deposit paying 7%, except that the relative wants to be repaid in a different currency. The 6 points look like easy income until the currency moves. A carry trade earns the interest gap only if the currency you hold does not fall by more than about that gap over the holding period. The rate gap is the cushion; the exchange rate is the risk.
Per Rs 100 of yen borrowed, the trade makes Rs 6 if the rupee holds its value, breaks even at a 5.6% fall, already loses Rs 0.42 at a 6% fall, and loses Rs 4.70 at a 10% fall.
Why is the cushion 5.6% and not 6%?
Work in yen per unit borrowed. At the end you hold 1.07 of rupee deposit, measured at today's exchange rate, and you owe 1.01 of yen; the rupee can lose value until 1.07 x (1 - d) = 1.01, so d = 5.6%. The fall applies to the whole 1.07, including the 0.07 of interest, which is why the cushion is a little smaller than 6. Borrow Rs 100 crore worth of yen and a 5.6% fall leaves you exactly even; a 10% fall loses about Rs 4.7 crore.
The relationship
$1.07\,(1 - d) = 1.01 \;\Rightarrow\; d = 1 - \frac{1.01}{1.07} = 5.6\%$
1.07 what each unit becomes in the rupee deposit at 7%
1.01 what you owe on each unit of yen borrowed at 1%
d the fall in the rupee's value in yen over the year
What it says in wordsThe trade breaks even when the grown rupee deposit, after the currency fall, buys back exactly the yen you owe.
What does the forward market say about this?
Covered interest parityThe rule that the forward exchange rate must offset the interest rate gap between two currencies, or else borrowing in one, lending in the other and locking in the forward would be an arbitrage. sets the one-year forward so that the rupee buys about 5.6% fewer yen forward than today, exactly the fall that wipes out the carry. So an unhedged carry trade is a bet that the rupee will fall by less than the forward already implies. The limitation to state: carry trades tend to earn small amounts steadily and then lose sharply when many holders unwind at once, so the spread of outcomes has a fat left tail that the breakeven alone does not show.
Where candidates lose it
The quick answer is 6%, the gap between the two rates. It misses that the currency loss applies to the interest earned as well as to the principal, so the true cushion is slightly smaller, and at exactly a 6% fall the trade is already losing.
The second loss is treating the 6-point gap as a return and forgetting the currency altogether. Name the exchange rate as the risk before doing any arithmetic.
What the interviewer asks next
The rupee weakens 10% against the yen over the year. What is the loss on Rs 100 crore of yen borrowed?
What one-year forward rate does covered interest parity imply, and what does it say about the trade?
Why do carry trades tend to lose money suddenly rather than gradually?

D_s	spread duration, 6
C	convexity, 50
\Delta s	the change in spread as a decimal, 50 basis points = 0.005

0.05	the starting weight, 5% of NAV
1.5	the price relative, 150 over 100
0.05 x 0.5	the loss as a share of starting NAV, 2.5%

40	revenue billed in rupees, unchanged
60 x 88/80	dollar revenue restated at the new rate, Rs 66
85	costs, all in rupees, unchanged

1,000	market value, Rs crore
300	net cash, Rs crore
0.06 x 300	the after-tax income the cash earns, Rs 18 crore

Try it first

What do duration and convexity each measure?

When does the convexity term start to matter?

Where candidates lose it

What the interviewer asks next

Try it first

How do the loss and the new weight work out?

Why is a losing short more dangerous than a losing long?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a weaker rupee help this company so much?

How do you size the profit move quickly?

What would you check before trusting the new margin?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the headline P/E misleading here?

How do you work it?

When would you not strip out all the cash?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the multiple one over a spread?

How do you work it out quickly?

What does this teach beyond the arithmetic?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the carry trade's income come from?

Why is the cushion 5.6% and not 6%?

What does the forward market say about this?

Where candidates lose it

What the interviewer asks next